In the recording, production and playback of audio, one important and widely used tool is equalization, the manipulation of signal level and phase as a function of frequency. Equalization may be used to correct problems in a recorded signal and for artistic purposes. Different genres of music can have characteristic power spectra, and equalization may be applied to program material so as to achieve the expected long-term power spectrum. In playback of audio, equalization may be used to compensate for resonances of a performance or listening space.
A common equalizer used for live sound and consumer playback is a so-called graphic equalizer, in which control is provided over the gain in each of a set of frequency bands. In a traditional graphic equalizer for audio shown in FIG. 1, the transfer function is controlled by specifying the gains for each of a set of cascaded shelving and peaking filters 104, 105 . . . 106, 107. See, e.g., Dennis A. Bohn, “Constant-Q graphic equalizers,” Journal of the Audio Engineering Society, 34:611-626, 1986. While it is desired that the transfer function magnitude smoothly interpolate the given gains, this is not always the case. As seen in the example of FIG. 3, if the filter bandwidths are small, the equalizer transfer function will exhibit ripples 302, tending towards unity gain at frequencies between the band centers. In the figure, the transfer function magnitudes of a set of peaking and shelving filters comprising the bands of a graphic equalizer are shown as dashed lines along with the transfer function magnitude of their cascade 302. The peaking and shelving filter gains are set according to input gains, shown as ‘o’ marks. On the other hand, as shown in FIG. 2, if the component filter bandwidths are sufficiently broad that the transfer function magnitude is smooth, the transfer function will often overshoot the desired gain due to contributions from adjacent bands.
This difficulty is well known, and in Justin Baird, Bruce Jackson and David McGrath, “Raised Cosine Equalization Utilizing Log Scale Filter Synthesis,” Audio Engineering Society 115th Convention, preprint 6257, San Francisco Calif., October 2004, Baird, et al. proposed making the band filters so-called mesa filters, rather than second-order sections, as is typical. Mesa filters have a prescribed band gain, crossfading to a gain of one outside the band. The crossfade approximates a raised cosine on a log-magnitude scale, and, as such, adjacent bands may be independently moved, with the system transfer function smoothly interpolating the band gains. The drawback is that the mesa filters are each made of seven parametric sections, and are costly to implement.
In another prior art approach proposed by Azizi (see Seyed-Ali Azizi, “A New Concept of Interference Compensation for Parametric and Graphic Equalizer Banks,” Audio Engineering Society 111th Convention, preprint 5482, New York N.Y., September 2001 and Seyed-Ali Azizi, “A New Concept of Interference Compensation for Parametric and Graphic Equalizer Banks,” Audio Engineering Society 111th Convention, preprint 5629, Munich Germany, May 2002), a correction filter 404 is added to the output of the equalizer, as shown in FIG. 4. The idea is that the correction filter is adjusted so that the cascade of the standard equalizer and correction approximately interpolates the desired band gains. Again the drawback is increased computational cost.
Another prior art method proposed by Azizi in the cited references is a filter design method, where the parameters describing the center frequencies, bandwidths and gains are adjusted in an iterative constrained nonlinear optimization process so as to achieve the desired band gains. Drawbacks to this approach include the computational cost of the optimization which Azizi describes as not suitable for real-time use, and the more serious difficulty that the iteration might get stuck in a local minimum.
Other filter design methods, such as Prony or Hankel methods (see Julius O. Smith III, Techniques for Digital Filter Design and System Identification with Application to the Violin, Ph.D. thesis, Stanford University, 1983), can be used to closely match a given transfer function magnitude. They, however, are not easily adapted to psychoacoustically meaningful goodness-of-fit measures, which involve minimizing dB differences in transfer function magnitude over a Bark or ERB frequency scaling. Those methods that apply psychoacoustic measures in designing filters can be computationally cumbersome due to the nonlinear optimization involved.
In any event, these design approaches are generally not useful for applications such as HRTF filtering (see E. M. Wenzel, “Localization in virtual acoustic displays,” Presence, 1:80-107, 1992), where the resulting filter needs to be slewed or interpolated between tabulated designs. The reason is that the poles and zeros maximizing a goodness-of-fit rarely can be related to particular features in the desired transfer function magnitude. As a result, there is often no clear way to process sets of tabulated filter coefficients that leads to a meaningful filter intermediate between table entries.
There remains a need in the art, therefore, to develop a graphic equalizer which interpolates the prescribed band gains, is computationally efficient to implement, and is parameterized in such a way that it may be interpolated or slewed between tabulated designs.